Dormand-Prince 4(5)

Description

Called by xcos, Dormand-Prince is a numerical solver providing an efficient fixed-size step method to solve Initial Value Problems of the form:

CVode and IDA use variable-size steps for the integration.

A drawback of that is the unpredictable computation time. With Dormand-Prince, we do not adapt to the complexity of the problem, but we guarantee a stable computation time.

As of now, this method is explicit, so it is not concerned with Newton or Functional iterations, and not advised for stiff problems.

It is an enhancement of the Euler method, which approximates
yn+1
by truncating the Taylor expansion.

By convention, to use fixed-size steps, the program first computes a fitting h that approaches the simulation parameter max step size.

An important difference of Dormand-Prince with the previous methods is that it computes up to the seventh derivative of y, while the others only use linear combinations of y and y'.

Here, the next value is determined by the present value
yn
plus the weighted average of six increments, where each increment is the product of the size of the interval, h, and an estimated slope specified by the function f(t,y):

k1 is the increment based on the slope at the beginning of the interval, using
yn
(Euler's method),

k2, k3, k4 and k5 are the increments based on the slope at respectively 0.2, 0.3, 0.8 and 0.9 of the interval, using combinations of each other,

k6 is the increment based on the slope at the end, also using combinations of the other ki.

We can see that with the ki, we progress in the derivatives of
yn
. In the computation of the ki, we deliberately use coefficients that yield an error in
O(h5)
at every step.

So the total error is
number of steps * O(h5)
. And since number of steps = interval size / h by definition, the total error is in
O(h4)
.

Althought the solver works fine for max step size up to
10-3
, rounding errors sometimes come into play as it approaches
4*10-4
. Indeed, the interval splitting cannot be done properly and we get capricious results.

Examples

The integral block returns its continuous state, we can evaluate it with Dormand-Prince by running the example:

// Import the diagram and set the ending timeloadScicos();loadXcosLibs();importXcosDiagram("SCI/modules/xcos/examples/solvers/ODE_Example.xcos");scs_m.props.tf=5000;// Select the Dormand-Prince solver and set the precisionscs_m.props.tol(6)=5;scs_m.props.tol(7)=10^-2;// Start the timer, launch the simulation and display timetic();tryxcos_simulate(scs_m,4);catchdisp(lasterror());endt=toc();disp(t,"Time for Dormand-Prince:");

The Scilab console displays:

Time for Dormand-Prince:
9.197

Now, in the following script, we compare the time difference between Dormand-Prince and CVode by running the example with the five solvers in turn:
Open the script

Time for BDF / Newton:
18.894
Time for BDF / Functional:
18.382
Time for Adams / Newton:
10.368
Time for Adams / Functional:
9.815
Time for Dormand-Prince:
7.842

These results show that on a nonstiff problem, for relatively same precision required and forcing the same step size, Dormand-Prince's computational overhead (compared to Runge-Kutta) is significant and is close to Adams/Functional. Its error to the solution is althought much smaller than the regular Runge-Kutta 4(5), for a small overhead in time.

Variable-size step ODE solvers are not appropriate for deterministic real-time applications because the computational overhead of taking a time step varies over the course of an application.